Question

We denote {0, 1}^{n} by sequences of 0’s and 1’s of
length n. Show that it is possible to order elements of {0,
1}^{n} so that two consecutive strings are different only
in one position

Answer #1

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

Find a recurrence relation for the number of bit sequences of
length n with an even number of 0s. please give me an initial
case.
+ (my question)
Let An is denote the number of bit sequences of length n with an
even number of 0s. A(1) = 1 because of "0" not "1"? A(2) = 2 but
why? why only "11" and "00" are acceptable for this problem?
"11,01,10,00" doesn't make sense?

We denote |S| the number of elements of a set S. (1) Let A and B
be two finite sets. Show that if A ∩ B = ∅ then A ∪ B is finite and
|A ∪ B| = |A| + |B| . Hint: Given two bijections f : A → N|A| and g
: B → N|B| , you may consider for instance the function h : A ∪ B →
N|A|+|B| defined as h (a) = f (a)...

1. Show work
Let a be the sequences defined by
an = ( – 2 )^(n+1)
a. Which term is greater, a 7, or a 8
b. Given any 2 consecutive terms, how can you
tell which one will yield the greater term of
the sequence ?

1. [10 marks] We begin with some mathematics regarding
uncountability. Let N = {0, 1, 2, 3, . . .} denote the set of
natural numbers.
(a) [5 marks] Prove that the set of binary numbers has the same
size as N by giving a bijection between the binary numbers and
N.
(b) [5 marks] Let B denote the set of all infinite sequences
over the English alphabet. Show that B is uncountable using a proof
by diagonalization.

Suppose we toss a fair coin n = 1 million times and write down
the outcomes: it gives a Heads-and-Tails-sequence of length n. Then
we call an integer i unique, if the i, i + 1, i + 2, i + 3, . . . ,
i + 18th elements of the sequence are all Heads. That is, we have a
block of 19 consecutive Heads starting with the ith element of the
sequence. Let Y denote the number of...

Consider sequences of n numbers, each in the set {1, 2, . . . ,
6}
(a) How many sequences are there if each number in the sequence
is distinct?
(b) How many sequences are there if no two consecutive numbers
are equal
(c) How many sequences are there if 1 appears exactly i times in
the sequence?

Two inﬁnite sequences {an}∞ n=0 and {bn}∞
n=0 satisfy the recurrence relations an+1 =
an −bn and bn+1 = 3an +
5bn for all n ≥ 0. Imitate the techniques
used to solve diﬀerential equations to ﬁnd general formulas for
an and bn in terms of n.

(1) How many bitstrings of length 8 begin with two 1’s or end
with three 1’s?
(2) How many bitstrings of length 10 contain three consecutive
0’s or 4 consecutive 1’s?

Prove n+1 < n for n>0 Assume for a value k; K+1< K We
now do the inductive hypothesis, by adding 1 to each side K+1+1
< k+1 => K+2< k+1 Thus we show that for all consecutive
integers k; k+1> k Where did we go wrong?

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